Mechanics Physics 151 Lecture 3 Lagrange’s Equations (Goldstein Chapter 1) Hamilton’s Principle (Chapter 2)
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MechanicsPhysics 151
Lecture 3Lagrange’s Equations
(Goldstein Chapter 1)
Hamilton’s Principle(Chapter 2)
What We Did Last Time
! Discussed multi-particle systems! Internal and external forces
! Laws of action and reaction
! Introduced constraints! Generalized coordinates
! Introduced Lagrange’s Equations! ... and didn’t do the derivation
" Let’s pick it up and start from there
Today’s Goals
! Derive Lagrange’s Eqn from Newton’s Eqn! Use D’Alembert’s principle! There will be a few assumptions
! Will make them clear as we go
! Introduce Hamilton’s Principle! Equivalent to Lagrange’s Equations
! Which in turn is equivalent to Newton’s Equations! Does not depend on coordinates by construction! Derivation in the next lecture
Lagrange’s Equations
! Express L = T – V in terms of generalized coordinates, their time-derivatives , and time t
! The potential V = V(q, t) must exist! i.e. all forces must be conservative
0j j
d L Ldt q q ∂ ∂− = ∂ ∂ !
( , , )L q q t T V≡ −!Kinetic energy
Potential energyLagrangian
Recipe
{ }jq { }jq!
Virtual Displacement
! Consider a system with constraints! Ordinary coordinates ri (i = 1...N)! Generalized coordinates qj (j = 1...n)
! Imagine moving all the particlesslightly
! Note that δri must satisfy the constraints
1 1 1 2
2 2 1 2
1 2
( , ,..., , )( , ,..., , )
( , ,..., , )
n
n
N N n
q q q tq q q t
q q q t
= = =
r rr r
r r"
i i iδ→ +r r r
Virtual displacement
ii j
j j
δ δ∂=∂∑ rr
3N coordinates not independent
n coordinates independent
j j jq q qδ→ +
! From Newton’s Equation of Motion
! Part of the force Fi must be due to constraints
! Applied force is “known”
! Constraint force fi (usually) does no work! Movement is perpendicular to the force! Exception: friction
! Now multiply by δri and sum over i
( )ai i i= +F F f
D’Alembert’s Principle
i i=F p! 0i i− =F p!
( ) ( )1 2( , ,..., ,..., , )a a
i i i N t=F F r r r r
“applied” force “constraint” force
0i iδ =f r
( ) 0ai i i+ − =F f p!
D’Alembert’s Principle
! Force of constraints dropped out because! Called D’Alembert’s Principle (1743)
! Now we switch from ri to qj
! Unit of Qj not always [force]! Qj qj is always [work]
( )( ) 0ai i i
i
δ− =∑ F p r!
0i iδ =f r
1st term ii j j j
i j jj
q Q qq
δ δ∂= =∂∑ ∑ ∑rF i
j ii j
∂≡∂∑ rF
Generalized force
“constraint” force is out of the game.You can forget (a)
D’Alembert’s Principle
! A bit of work can show
! D’Alembert’s Principle becomes
,2nd term i i
i i i j i i ji i j i jj j
q m qq q
δ δ δ∂ ∂= = =∂ ∂∑ ∑ ∑ ∑r rp r p r! ! !!
2 2
2 2i i i
ij j j
v vdq dt q q
∂ ∂ ∂→ − ∂ ∂ ∂
rr!!!
jj j j
d T T qdt q q
δ ∂ ∂ = − ∂ ∂
∑ !
0j jj j j
d T T Q qdt q q
δ ∂ ∂ − − = ∂ ∂
∑ !
2
2i
i
mvT ≡∑
Lagrange’s Equations
! Generalized coordinates qj are independent
! Assume forces are conservative
0j jj j j
d T T Q qdt q q
δ ∂ ∂ − − = ∂ ∂
∑ !These are free
jj j
d T T Qdt q q ∂ ∂− = ∂ ∂ !
Almost there!
i iV= −∇F
i ij i i
i ij j j
VQ Vq q q
∂ ∂ ∂≡ = − ∇ = −∂ ∂ ∂∑ ∑r rF
Throw thisback in
Lagrange’s Equations
! Assume that V does not depend on
( ) 0j j
T Vd Tdt q q ∂ −∂ − = ∂ ∂ !
jq! 0j
Vq
∂ =∂ !
0j j
d L Ldt q q ∂ ∂− = ∂ ∂ !
Finally ( , , ) ( , )j j jL T q q t V q t= −!
Done!
Assumptions We Made
! Constraints are holonomic! We always assume this
! Constraint forces do no work! Forget frictions
! Applied forces are conservative! Lagrange’s Eqn. itself is OK if V depends explicitly on t
! Potential V does not depend on
1 2( , ,..., , )i i nq q q t=r r
0i iδ =f r
i iV= −∇F
jq! 0j
Vq
∂ =∂ !
Will review the last assumption later
Example: Time-Dependent
! Transformation functions may depend on t! Generalized coordinate system may move! E.g. coordinate system fixed to the Earth
! An example
( , )i i jq t=r r
mass m on a rail
spring constant Knatural length l
angular velocity αl + r
Example: Time-Dependent
! Transformation functions:
! Kinetic energy
! Potential energy
( ) cos( )sin
x l r ty l r t
αα
= + = +
{ } { }2 2 2 2 2( )2 2m mT x y r l r α= + = + +! ! !
2
2KV r=
{ }2 2 2 2( )2 2m KL r l r rα= + + −!
2 ( ) 0d L L mr m l r Krdt r r
α∂ ∂ − = − + + = ∂ ∂ !!
!Lagrange’s Equation
Example: Time-Dependent
! If K > mα2, a harmonic oscillator with! Center of oscillation is shifted by
! If K < mα2, moves away exponentially! If K = mα2, velocity is constant
! Centripetal force balances with the spring force
2 ( ) 0d L L mr m l r Krdt r r
α∂ ∂ − = − + + = ∂ ∂ !!
!2
22( ) 0m lmr K m r
K mαα
α
+ − − = − !!
2K mm
αω −=
Note on Arbitrarity
! Lagrangian is not unique for a given system! If a Lagrangian L describes a system
! One can prove
( , )dF q tL Ldt
′ = + works as well for any function F
0d dF dFdt q dt q dt ∂ ∂ − = ∂ ∂ !
dF F Fqdt q t
∂ ∂= +∂ ∂!using
Assumptions We Made
! Constraints are holonomic! We always assume this
! Constraint forces do no work! Forget frictions
! Applied forces are conservative! Lagrange’s Eqn. itself is OK if V depends explicitly on t
! Potential V does not depend on
1 2( , ,..., , )i i nq q q t=r r
0i iδ =f r
i iV= −∇F
jq! 0j
Vq
∂ =∂ !
Let’s review the last assumption
Velocity-Dependent Potential
! We assumed and so that
! We could do the same if we had
0j
Vq
∂ =∂ !
jj j
d T T Qdt q q ∂ ∂− = ∂ ∂ !
( ) ( ) 0j j
d T V T Vdt q q ∂ − ∂ −− = ∂ ∂ !
This had to be 0
jj j
U d UQq dt q
∂ ∂= − + ∂ ∂ !( , , )j jU U q q t= !
Generalized, or velocity-dependent“potential”
( , , ) ( , , )j j j jL T q q t U q q t= −! !
jj
VQq
∂= −∂
EM Force on Particle
! Lorentz force on a charged particle
! E and B fields are given by
! Force is v-dependent " Need a v-dependent potential
! Lagrangian is
[ ( )]q= + ×F E v B
tφ ∂= −∇ −
∂AE = ∇×B A
U q qφ= − ⋅A v works check
212
L mv q qφ= − + ⋅A v
Velocity-dependent. Can’t find a usual
potential V
Physics 15b
Monogenic System
! If all forces in a system are derived from a generalized potential,its called a monogenic system! U is a function of! Lorentz force is monogenic
! A monogenic system is conservative only if
! Or
! Lagrange’s Equation works on a monogenic system
jj j
U d UQq dt q
∂ ∂= − + ∂ ∂ !, ,q q t!
( )U U q=
0U Uq t
∂ ∂= =∂ ∂!
Hamilton’s Principle
! We derived Lagrange’s Eqn from Newton’s Eqn using a “differential principle”! D’Alembert’s principle uses infinitesimal displacements
! It’s possible to do it with an “integral principle”
Hamilton’s Principle
Configuration Space
! Generalized coordinates q1,...,qn fully describe the system’s configuration at any moment
! Imagine an n-dimensional space! Each point in this space (q1,...,qn)
corresponds to one configuration of the system! Time evolution of the system " A curve in the
configuration space
configurationspace
real space configuration space
Action Integral
! A system is moving as! Lagrangian is
! Action I depends on the entire path from t1 to t2
! Choice of coordinates qj does not matter! Action is invariant under coordinate transformation
( ) 1...j jq q t j n= =
( , , ) ( ( ), ( ), )L q q t L q t q t t=! !
integrate2
1
t
tI Ldt= ∫ Action, or action integral
Hamilton’s Principle
! This is equivalent to Lagrange’s Equations! We will prove this
! Three equivalent formulations! Newton’s Eqn depends explicitly on x-y-z coordinates! Lagrange’s Eqn is same for any generalized coordinates! Hamilton’s Principle refers to no coordinates
! Everything is in the action integral
The action integral of a physical system is stationaryfor the actual path
We will also define “stationary”
Hamilton’s Principle is more fundamental probably...
Stationary
! Consider two paths that are close to each other! Difference is infinitesimal
! Stationary means that thedifference of the action integrals iszero to the 1st order of δq(t)! Similar to “first derivative = 0”
! Almost same as saying “minimum”! It could as well be maximum
configuration space
1t
2t
( )q t
( ) ( )q t q tδ+2 2
1 1
( , , ) ( , , ) 0t t
t tI L q q q q t dt L q q t dtδ δ δ= + + − =∫ ∫! ! !
1 2( ) ( ) 0q t q tδ δ= =
Infinitesimal Path Difference
! What’s δq(t)?! It’s arbitrary … sort of! It has to be zero at t1 and t2
! It’s well-behaving
! Have to shrink it to zero! Trick: write it as
! α is a parameter, which we’ll make " 0! η(t) is an arbitrary well-behaving function
configuration space
1t
2t
( )q t
( ) ( )q t q tδ+Continuous, non-singular,continuous 1st and 2nd derivatives
( ) ( )q t tδ αη=
1 2( ) ( ) 0t tη η= =
Don’t worry too much
Hamilton " Lagrange
! To derive Lagrange’s Eqns from Hamilton’s Principle
! Define
! δI is then
! We must show that leads to Lagrange’s Eqns
2
1
( ) ( ( ) ( ), ( ) ( ), )t
tI L q t t q t t t dtα αη αη≡ + +∫ ! !
2 2
1 1
( , , ) ( , , ) 0t t
t tI L q q q q t dt L q q t dtδ δ δ= + + − =∫ ∫! ! !
[ ]0
lim ( ) (0)I Iα
α→
−0
I dα
αα =
∂ ∂
0
0I
αα =
∂ = ∂
A bit of work. Will do it on Thursday
Summary
! Derived Lagrange’s Eqn from Newton’s Eqn! Using D’Alembert’s Principle Differential approach
! Assumptions we made:! Constraints are holonomic " Generalized coordinates! Forces of constraints do no work " No frictions! Other forces are monogenic " Generalized potential
! Introduced Hamilton’s Principle! Integral approach! Defined the action integral and “stationary”! Derivation in the next lecture
jj j
U d UQq dt q
∂ ∂= − + ∂ ∂ !
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